being \(\hat{K}^{+}=\frac{\hat{a}^{\dagger 2}}{2}\) , \( \hat{K}^{0}=\frac{\left(\hat{n} + 1/2\right)}{2}\)  and \( \hat{K}^{-}=\frac{\hat{a}^{2}}{2}\) . The commutation relations of these operators are \(\left[\hat{K}^{+}, \hat{K}^{-} \right]=-2\hat{K}^0\) , \(\left[\hat{K}^{0}, \hat{K}^{\pm } \right]=\pm \hat{K}^{\pm }\) , and they are the generators of the su(1,1) algebra [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}.
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